library(EBImage)
tau <<- 2*pi
# Size in pixels of final image:
N = 320
M = 640
# At full letter_width = 1, letters might look
# too close to each other, especially if they
# have vertical strokes right next to each other.
letter_width = 1
letter_width = min(1, letter_width)
letter_width = max(0.1, letter_width)
fractal_str = "FRACTAL"
fractal_str = toupper(fractal_str)
max_iter <<- ceiling(log(M) / log(nchar(fractal_str)))
# Matrix to hold the image:
fractal_img <<- matrix(rep(0, N*M), ncol=M, nrow=N)
# Discretisation of the parametric curves defining the strokes,
# with t in [0, 1].
delta_t <<- 0.01
letter_a = function() {
# Returns a list of parametric curves for the letter A.
t = seq(0, 1, by=delta_t)
n = length(t)
# Stroke 1:
x1_1 = 0.02 + 0*t
y1_1 = 1 - 0.9*t
x1_2 = 0.12 + 0*t
y1_2 = 1 - 0.9*t
# Stroke 2:
x2_1 = seq(0.02, 0.98, length.out = n)
y2_1 = 0*t
x2_2 = x2_1
y2_2 = 0.1 + 0*t
# Stroke 3:
x3_1 = 0.98 + 0*t
y3_1 = 0.1 + 0.9*t
x3_2 = 0.88 + 0*t
y3_2 = 0.1 + 0.9*t
# Stroke 4:
x4_1 = seq(0.12, 0.88, length.out = n)
y4_1 = 0.4 + 0*t
x4_2 = x4_1
y4_2 = 0.5 + 0*t
return(list(list(x1 = x1_1, y1 = y1_1, x2 = x1_2, y2 = y1_2),
list(x1 = x2_1, y1 = y2_1, x2 = x2_2, y2 = y2_2),
list(x1 = x3_1, y1 = y3_1, x2 = x3_2, y2 = y3_2),
list(x1 = x4_1, y1 = y4_1, x2 = x4_2, y2 = y4_2)))
}
letter_b = function() {
# Returns a list of parametric curves for the letter B.
t = seq(0, 1, by=delta_t)
# Stroke 1:
x1_1 = 0.02 + 0*t
y1_1 = 1 - t
x1_2 = 0.12 + 0*t
y1_2 = 1 - t
# Stroke 2:
t = seq(-tau/4, tau/4, length.out = length(t))
x2_1 = 0.12 + 0.86*cos(t)
y2_1 = 0.75 + 0.25*sin(t)
x2_2 = 0.12 + 0.76*cos(t)
y2_2 = 0.75 + 0.15*sin(t)
# Stroke 3:
x3_1 = 0.12 + 0.86*cos(t)
y3_1 = 0.25 + 0.25*sin(t)
x3_2 = 0.12 + 0.76*cos(t)
y3_2 = 0.25 + 0.15*sin(t)
return(list(list(x1 = x1_1, y1 = y1_1, x2 = x1_2, y2 = y1_2),
list(x1 = x2_1, y1 = y2_1, x2 = x2_2, y2 = y2_2),
list(x1 = x3_1, y1 = y3_1, x2 = x3_2, y2 = y3_2)))
}
letter_c = function() {
t = seq(0, 1, by=delta_t)
t_max = 0.8*tau
t_min = 0.2*tau
t = seq(t_max, t_min, length.out = length(t))
# (x, y) = (a*cos(t), b*sin(t))
a = (0.98 - 0.02) / (cos(t_min) + 1)
x0 = a + 0.02
x1_1 = x0 + a*cos(t)
y1_1 = 0.5 - 0.5*sin(t)
x1_2= x0 + (a - 0.1)*cos(t)
y1_2 = 0.5 - 0.4*sin(t)
return(list(list(x1 = x1_1, y1 = y1_1, x2 = x1_2, y2 = y1_2)))
}
letter_d = function() {
t = seq(0, 1, by=delta_t)
# Stroke 1:
x1_1 = 0.02 + 0*t
y1_1 = 1 - t
x1_2 = 0.12 + 0*t
y1_2 = 1 - t
# Stroke 2:
t = seq(-tau/4, tau/4, length.out = length(t))
x2_1 = 0.12 + 0.86*cos(t)
y2_1 = 0.5 + 0.5*sin(t)
x2_2 = 0.12 + 0.76*cos(t)
y2_2 = 0.5 + 0.4*sin(t)
return(list(list(x1 = x1_1, y1 = y1_1, x2 = x1_2, y2 = y1_2),
list(x1 = x2_1, y1 = y2_1, x2 = x2_2, y2 = y2_2)))
}
letter_e = function() {
# Returns a list of parametric curves for the letter E.
t = seq(0, 1, by=delta_t)
# Stroke 1:
x1_1 = 0.02 + 0*t
y1_1 = 1 - t
x1_2 = 0.12 + 0*t
y1_2 = 1 - t
# Stroke 2:
x2_1 = 0.12 + (0.98 - 0.12)*t
y2_1 = 0 + 0*t
x2_2 = 0.12 + (0.98 - 0.12)*t
y2_2 = 0.1 + 0*t
# Stroke 3:
x3_1 = 0.12 + (0.98 - 0.12)*t
y3_1 = 0.45 + 0*t
x3_2 = 0.12 + (0.98 - 0.12)*t
y3_2 = 0.55 + 0*t
# Stroke 4:
x4_1 = 0.12 + (0.98 - 0.12)*t
y4_1 = 0.9 + 0*t
x4_2 = 0.12 + (0.98 - 0.12)*t
y4_2 = 1 + 0*t
return(list(list(x1 = x1_1, y1 = y1_1, x2 = x1_2, y2 = y1_2),
list(x1 = x2_1, y1 = y2_1, x2 = x2_2, y2 = y2_2),
list(x1 = x3_1, y1 = y3_1, x2 = x3_2, y2 = y3_2),
list(x1 = x4_1, y1 = y4_1, x2 = x4_2, y2 = y4_2)))
}
letter_f = function() {
# Returns a list of parametric curves for the letter E.
t = seq(0, 1, by=delta_t)
# Stroke 1:
x1_1 = 0.02 + 0*t
y1_1 = 1 - t
x1_2 = 0.12 + 0*t
y1_2 = 1 - t
# Stroke 2:
x2_1 = 0.12 + (0.98 - 0.12)*t
y2_1 = 0 + 0*t
x2_2 = 0.12 + (0.98 - 0.12)*t
y2_2 = 0.1 + 0*t
# Stroke 3:
x3_1 = 0.12 + (0.98 - 0.12)*t
y3_1 = 0.45 + 0*t
x3_2 = 0.12 + (0.98 - 0.12)*t
y3_2 = 0.55 + 0*t
return(list(list(x1 = x1_1, y1 = y1_1, x2 = x1_2, y2 = y1_2),
list(x1 = x2_1, y1 = y2_1, x2 = x2_2, y2 = y2_2),
list(x1 = x3_1, y1 = y3_1, x2 = x3_2, y2 = y3_2)))
}
letter_g = function() {
t = seq(0, 1, by=delta_t)
t_min = 0.2*tau
# (x, y) = (a*cos(t), b*sin(t))
a = (0.98 - 0.02) / (cos(t_min) + 1)
x0 = a + 0.02
t_max = tau - acos((0.88 - x0)/a)
print(t_max/tau)
t = seq(t_max, t_min, length.out = length(t))
# Stroke 1:
x1_1 = x0 + a*cos(t)
y1_1 = 0.5 - 0.5*sin(t)
x1_2 = x0 + (a - 0.1)*cos(t)
y1_2 = 0.5 - 0.4*sin(t)
# Stroke 2:
t = seq(0, 1, by=delta_t)
x2_1 = 0.88 + 0*t
y2_1 = 1 - 0.45*t
x2_2 = 0.98 - 0*t
y2_2 = 1 - 0.45*t
return(list(list(x1 = x1_1, y1 = y1_1, x2 = x1_2, y2 = y1_2),
list(x1 = x2_1, y1 = y2_1, x2 = x2_2, y2 = y2_2)))
}
letter_h = function() {
# Returns a list of parametric curves for the letter H.
t = seq(0, 1, by=delta_t)
# Stroke 1:
x1_1 = 0.02 + 0*t
y1_1 = 1 - t
x1_2 = 0.12 + 0*t
y1_2 = 1 - t
# Stroke 2:
x2_1 = 0.12 + (0.88 - 0.12)*t
y2_1 = 0.45 + 0*t
x2_2 = 0.12 + (0.88 - 0.12)*t
y2_2 = 0.55 + 0*t
# Stroke 3:
x3_1 = 0.88 + 0*t
y3_1 = 1 - t
x3_2 = 0.98 + 0*t
y3_2 = 1 - t
return(list(list(x1 = x1_1, y1 = y1_1, x2 = x1_2, y2 = y1_2),
list(x1 = x2_1, y1 = y2_1, x2 = x2_2, y2 = y2_2),
list(x1 = x3_1, y1 = y3_1, x2 = x3_2, y2 = y3_2)))
}
letter_i = function() {
# Returns a list of parametric curves for the letter I.
t = seq(0, 1, by=delta_t)
# Stroke 1:
x1_1 = 0.45 + 0*t
y1_1 = 0.9 - 0.8*t
x1_2 = 0.55 + 0*t
y1_2 = 0.9 - 0.8*t
# Stroke 2:
x2_1 = 0.3 + 0.4*t
y2_1 = 0*t
x2_2 = 0.3 + 0.4*t
y2_2 = 0.1 + 0*t
# Stroke 3:
x3_1 = 0.3 + 0.4*t
y3_1 = 0.9 + 0*t
x3_2 = 0.3 + 0.4*t
y3_2 = 1 + 0*t
return(list(list(x1 = x1_1, y1 = y1_1, x2 = x1_2, y2 = y1_2),
list(x1 = x2_1, y1 = y2_1, x2 = x2_2, y2 = y2_2),
list(x1 = x3_1, y1 = y3_1, x2 = x3_2, y2 = y3_2)))
}
letter_j = function() {
t = seq(0, 1, by=delta_t)
t = seq(-tau/2, 0, length.out = length(t) - 1)
# Stroke 1:
x1_1 = c(0.5 + 0.15*cos(t), 0.65)
y1_1 = c(0.75 - 0.15*sin(t), 0.1)
x1_2 = c(0.5 + 0.25*cos(t), 0.75)
y1_2 = c(0.75 - 0.25*sin(t), 0.1)
# Stroke 2:
t = seq(0, 1, by=delta_t)
x2_1 = 0.5 + 0.4*t
y2_1 = 0 + 0*t
x2_2 = 0.5 + 0.4*t
y2_2 = 0.1 + 0*t
return(list(list(x1 = x1_1, y1 = y1_1, x2 = x1_2, y2 = y1_2),
list(x1 = x2_1, y1 = y2_1, x2 = x2_2, y2 = y2_2)))
}
letter_k = function() {
t = seq(0, 1, by=delta_t)
n = length(t)
# Stroke 1:
x1_1 = 0.02 + 0*t
y1_1 = 1 - t
x1_2 = 0.12 + 0*t
y1_2 = 1 - t
# Stroke 2:
x2_1 = seq(0.22, 0.98, length.out = n)
y2_1 = seq(0.5, 1, length.out = n)
x2_2 = seq(0.12, 0.88, length.out = n)
y2_2 = seq(0.5, 1, length.out = n)
# Stroke 3:
x3_1 = seq(0.12, 0.88, length.out = n)
y3_1 = seq(0.5, 0, length.out = n)
x3_2 = seq(0.22, 0.98, length.out = n)
y3_2 = seq(0.5, 0, length.out = n)
return(list(list(x1 = x1_1, y1 = y1_1, x2 = x1_2, y2 = y1_2),
list(x1 = x2_1, y1 = y2_1, x2 = x2_2, y2 = y2_2),
list(x1 = x3_1, y1 = y3_1, x2 = x3_2, y2 = y3_2)))
}
letter_l = function() {
t = seq(0, 1, by=delta_t)
# Stroke 1:
x1_1 = 0.02 + 0*t
y1_1 = 1 - t
x1_2 = 0.12 + 0*t
y1_2 = 1 - t
# Stroke 2:
x2_1 = seq(0.12, 0.98, length.out = length(t))
y2_1 = 0.9 + 0*t
x2_2 = x2_1
y2_2 = 1 + 0*t
return(list(list(x1 = x1_1, y1 = y1_1, x2 = x1_2, y2 = y1_2),
list(x1 = x2_1, y1 = y2_1, x2 = x2_2, y2 = y2_2)))
}
letter_m = function() {
t = seq(0, 1, by=delta_t)
n = length(t)
# Stroke 1:
x1_1 = 0.02 + 0*t
y1_1 = 1 - t
x1_2 = 0.12 + 0*t
y1_2 = 1 - t
# Stroke 2:
x2_1 = seq(0.12, 0.5, length.out = n)
y2_1 = seq(0, 0.4, length.out = n)
x2_2 = seq(0.12, 0.5, length.out = n)
y2_2 = seq(0.1, 0.5, length.out = n)
# Stroke 3:
x3_1 = seq(0.5, 0.88, length.out = n)
y3_1 = seq(0.4, 0, length.out = n)
x3_2 = seq(0.5, 0.88, length.out = n)
y3_2 = seq(0.5, 0.1, length.out = n)
# Stroke 4:
x4_1 = 0.98 + 0*t
y4_1 = t
x4_2 = 0.88 + 0*t
y4_2 = t
return(list(list(x1 = x1_1, y1 = y1_1, x2 = x1_2, y2 = y1_2),
list(x1 = x2_1, y1 = y2_1, x2 = x2_2, y2 = y2_2),
list(x1 = x3_1, y1 = y3_1, x2 = x3_2, y2 = y3_2),
list(x1 = x4_1, y1 = y4_1, x2 = x4_2, y2 = y4_2)))
}
letter_n = function() {
# Returns a list of parametric curves for the letter N.
t = seq(0, 1, by=delta_t)
n = length(t)
# Stroke 1:
x1_1 = 0.02 + 0*t
y1_1 = 1 - t
x1_2 = 0.12 + 0*t
y1_2 = 1 - t
# Stroke 2:
x2_1 = 0.12 + (0.88 - 0.12)*t
y2_1 = seq(0, 1 - 0.1*sqrt(2), length.out = n)
x2_2 = 0.12 + (0.88 - 0.12)*t
y2_2 = seq(0.1*sqrt(2), 1, length.out = n)
# Stroke 3:
x3_1 = 0.88 + 0*t
y3_1 = 1 - t
x3_2 = 0.98 + 0*t
y3_2 = 1 - t
return(list(list(x1 = x1_1, y1 = y1_1, x2 = x1_2, y2 = y1_2),
list(x1 = x2_1, y1 = y2_1, x2 = x2_2, y2 = y2_2),
list(x1 = x3_1, y1 = y3_1, x2 = x3_2, y2 = y3_2)))
}
letter_o = function() {
t = seq(0, 1, by=delta_t)
t = seq(tau/4, -tau/4, length.out = length(t))
# Stroke 1:
x1_1 = 0.5 + 0.48*cos(t)
y1_1 = 0.5 - 0.5*sin(t)
x1_2 = 0.5 + 0.38*cos(t)
y1_2 = 0.5 - 0.4*sin(t)
# Stroke 2:
x2_1 = 0.5 - 0.48*cos(t)
y2_1 = 0.5 + 0.5*sin(t)
x2_2 = 0.5 - 0.38*cos(t)
y2_2 = 0.5 + 0.4*sin(t)
return(list(list(x1 = x1_1, y1 = y1_1, x2 = x1_2, y2 = y1_2),
list(x1 = x2_1, y1 = y2_1, x2 = x2_2, y2 = y2_2)))
}
letter_p = function() {
# Returns a list of parametric curves for the letter P.
t = seq(0, 1, by=delta_t)
# Stroke 1:
x1_1 = 0.02 + 0*t
y1_1 = 1 - t
x1_2 = 0.12 + 0*t
y1_2 = 1 - t
# Stroke 2:
t = seq(-tau/4, tau/4, length.out = length(t))
x2_1 = 0.12 + 0.86*cos(t)
y2_1 = 0.25 + 0.25*sin(t)
x2_2 = 0.12 + 0.76*cos(t)
y2_2 = 0.25 + 0.15*sin(t)
return(list(list(x1 = x1_1, y1 = y1_1, x2 = x1_2, y2 = y1_2),
list(x1 = x2_1, y1 = y2_1, x2 = x2_2, y2 = y2_2)))
}
q_stroke_t = function(a, b, m, c) {
# Ellipse for the main part of the letter Q
# is defined by (x, y) = (a*cos(t), b*sin(t))
#
# The straight stroke is a line y = m*x + c.
# This function returns the intersection time t.
t0 = 0.15*tau
for (ct in 1:5) {
t0 = t0 - (a*m*cos(t0) - b*sin(t0) + c) / (-a*m*sin(t0) - b*cos(t0))
}
return(t0)
}
letter_q = function() {
# Returns a list of parametric curves for the letter P.
t = seq(0, 1, by=delta_t)
# Stroke 1:
x1_1 = seq(0.5 + 0.1/sqrt(2), 0.75 + 0.1/sqrt(2), length.out = length(t))
y1_1 = seq(0.75 - 0.1/sqrt(2), 1 - 0.1/sqrt(2), length.out = length(t))
x1_2 = seq(0.5, 0.75, length.out = length(t))
y1_2 = seq(0.75, 1, length.out = length(t))
# Stroke 2:
a = 0.48
b = 0.5
t0 = q_stroke_t(a, b, 1, 0.25)
t = seq(t0, 3*tau/4, length.out = length(t))
x2_1 = 0.5 + a*cos(t)
y2_1 = 0.5 + b*sin(t)
a = 0.38
b = 0.4
t0 = q_stroke_t(a, b, 1, 0.25)
t = seq(t0, 3*tau/4, length.out = length(t))
x2_2 = 0.5 + a*cos(t)
y2_2 = 0.5 + b*sin(t)
# Stroke 3:
a = 0.48
b = 0.5
t0 = q_stroke_t(a, b, 1, 0.25 - .1*sqrt(2))
t = seq(3*tau/4, tau+t0, length.out = length(t))
x3_1 = 0.5 + a*cos(t)
y3_1 = 0.5 + b*sin(t)
a = 0.38
b = 0.4
t0 = q_stroke_t(a, b, 1, 0.25 - .1*sqrt(2))
t = seq(3*tau/4, tau+t0, length.out = length(t))
x3_2 = 0.5 + a*cos(t)
y3_2 = 0.5 + b*sin(t)
return(list(list(x1 = x1_1, y1 = y1_1, x2 = x1_2, y2 = y1_2),
list(x1 = x2_1, y1 = y2_1, x2 = x2_2, y2 = y2_2),
list(x1 = x3_1, y1 = y3_1, x2 = x3_2, y2 = y3_2)))
}
letter_r = function() {
# Returns a list of parametric curves for the letter R.
t = seq(0, 1, by=delta_t)
# Stroke 1:
x1_1 = 0.02 + 0*t
y1_1 = 1 - t
x1_2 = 0.12 + 0*t
y1_2 = 1 - t
# Stroke 2:
t = seq(-tau/4, tau/4, length.out = length(t))
x2_1 = 0.12 + 0.86*cos(t)
y2_1 = 0.25 + 0.25*sin(t)
x2_2 = 0.12 + 0.76*cos(t)
y2_2 = 0.25 + 0.15*sin(t)
# Stroke 3:
t = seq(0, 1, by=delta_t)
x3_1 = seq(0.12 + 0.1*sqrt(2), 0.98, length.out = length(t))
y3_1 = 0.5 + 0.5*t
x3_2 = seq(0.12, 0.98 - 0.1*sqrt(2), length.out = length(t))
y3_2 = 0.5 + 0.5*t
return(list(list(x1 = x1_1, y1 = y1_1, x2 = x1_2, y2 = y1_2),
list(x1 = x2_1, y1 = y2_1, x2 = x2_2, y2 = y2_2),
list(x1 = x3_1, y1 = y3_1, x2 = x3_2, y2 = y3_2)))
}
letter_s = function() {
t = seq(0, 1, by=delta_t)
t = seq(tau/4, 3*tau/4, length.out = length(t) - 1)
# Stroke 1:
x1_1 = c(0.5 + 0.48*cos(t), 0.98)
y1_1 = c(0.275 + 0.275*sin(t), 0)
x1_2 = c(0.5 + 0.38*cos(t), 0.98)
y1_2 = c(0.275 + 0.175*sin(t), 0.1)
# Stroke 2:
t = seq(tau/4, -tau/4, length.out = length(t))
x2_1 = c(0.02, 0.5 + 0.38*cos(t))
y2_1 = c(0.9, 0.725 + 0.175*sin(t))
x2_2 = c(0.02, 0.5 + 0.48*cos(t))
y2_2 = c(1, 0.725 + 0.275*sin(t))
return(list(list(x1 = x1_1, y1 = y1_1, x2 = x1_2, y2 = y1_2),
list(x1 = x2_1, y1 = y2_1, x2 = x2_2, y2 = y2_2)))
}
letter_t = function() {
t = seq(0, 1, by=delta_t)
# Stroke 1:
x1_1 = 0.45 + 0*t
y1_1 = 1 - 0.9*t
x1_2 = 0.55 + 0*t
y1_2 = 1 - 0.9*t
# Stroke 2:
x2_1 = seq(0.02, 0.98, length.out = length(t))
y2_1 = 0*t
x2_2 = x2_1
y2_2 = 0.1 + 0*t
return(list(list(x1 = x1_1, y1 = y1_1, x2 = x1_2, y2 = y1_2),
list(x1 = x2_1, y1 = y2_1, x2 = x2_2, y2 = y2_2)))
}
letter_u = function() {
t = seq(0, 1, by=delta_t)
t = seq(tau/4, 0, length.out = length(t) - 1)
# Stroke 1:
x1_1 = c(0.5 + 0.38*cos(t), 0.88)
y1_1 = c(0.75 + 0.15*sin(t), 0)
x1_2 = c(0.5 + 0.48*cos(t), 0.98)
y1_2 = c(0.75 + 0.25*sin(t), 0)
# Stroke 2:
t = seq(0, tau/4, length.out = length(t))
x2_1 = c(0.12, 0.5 - 0.38*cos(t))
y2_1 = c(0, 0.75 + 0.15*sin(t))
x2_2 = c(0.02, 0.5 - 0.48*cos(t))
y2_2 = c(0, 0.75 + 0.25*sin(t))
return(list(list(x1 = x1_1, y1 = y1_1, x2 = x1_2, y2 = y1_2),
list(x1 = x2_1, y1 = y2_1, x2 = x2_2, y2 = y2_2)))
}
letter_v = function() {
t = seq(0, 1, by=delta_t)
n = length(t)
# Stroke 1:
x1_1 = seq(0.02, 0.5, length.out = n)
y1_1 = (1 - sqrt(2)*0.1)*t
x1_2 = x1_1
y1_2 = y1_1 + sqrt(2)*0.1
# Stroke 2:
x2_1 = seq(0.5, 0.98, length.out = n)
y2_1 = (1 - sqrt(2)*0.1)*(1 - t)
x2_2 = x2_1
y2_2 = y2_1 + sqrt(2)*0.1
return(list(list(x1 = x1_1, y1 = y1_1, x2 = x1_2, y2 = y1_2),
list(x1 = x2_1, y1 = y2_1, x2 = x2_2, y2 = y2_2)))
}
letter_w = function() {
t = seq(0, 1, by=delta_t)
n = length(t)
# Stroke 1:
x1_1 = 0.12 + 0*t
y1_1 = t
x1_2 = 0.02 + 0*t
y1_2 = t
# Stroke 2:
x2_1 = seq(0.12, 0.5, length.out = n)
y2_1 = seq(0.9, 0.5, length.out = n)
x2_2 = seq(0.12, 0.5, length.out = n)
y2_2 = seq(1, 0.6, length.out = n)
# Stroke 3:
x3_1 = seq(0.5, 0.88, length.out = n)
y3_1 = seq(0.5, 0.9, length.out = n)
x3_2 = seq(0.5, 0.88, length.out = n)
y3_2 = seq(0.6, 1, length.out = n)
# Stroke 4:
x4_1 = 0.88 + 0*t
y4_1 = 1 - t
x4_2 = 0.98 + 0*t
y4_2 = 1 - t
return(list(list(x1 = x1_1, y1 = y1_1, x2 = x1_2, y2 = y1_2),
list(x1 = x2_1, y1 = y2_1, x2 = x2_2, y2 = y2_2),
list(x1 = x3_1, y1 = y3_1, x2 = x3_2, y2 = y3_2),
list(x1 = x4_1, y1 = y4_1, x2 = x4_2, y2 = y4_2)))
}
letter_x = function() {
t = seq(0, 1, by=delta_t)
n = length(t)
# Stroke 1:
x1_1 = seq(0.02, 0.98, length.out = n)
y1_1 = seq(1 - 0.1*sqrt(2), 0, length.out = n)
x1_2 = x1_1
y1_2 = y1_1 + 0.1*sqrt(2)
# Stroke 2:
# The second and third strokes will be either side of the first stroke.
# Need to find the intersection points. 'a' for top of stroke, 'b' for
# bottom of stroke.
m1a = (0.1*sqrt(2) - 1) / 0.96
c1a = -m1a*0.98
m1b = m1a
c1b = c1a + 0.1*sqrt(2)
m2a = (1 - 0.1*sqrt(2)) / 0.96
c2a = -m2a*0.02
m2b = m2a
c2b = c2a + 0.1*sqrt(2)
x0 = (c2a - c1a) / (m1a - m2a)
y0 = m1a*x0 + c1a
x2_1 = seq(0.02, x0, length.out = n)
y2_1 = seq(0, y0, length.out = n)
x0 = (c2b - c1a) / (m1a - m2b)
y0 = m1a*x0 + c1a
x2_2 = seq(0.02, x0, length.out = n)
y2_2 = seq(0.1*sqrt(2), y0, length.out = n)
# Stroke 3:
x0 = (c2a - c1b) / (m1b - m2a)
y0 = m1b*x0 + c1b
x3_1 = seq(x0, 0.98, length.out = n)
y3_1 = seq(y0, 1 - 0.1*sqrt(2), length.out = n)
x0 = (c2b - c1b) / (m1b - m2b)
y0 = m1b*x0 + c1b
x3_2 = seq(x0, 0.98, length.out = n)
y3_2 = seq(y0, 1, length.out = n)
return(list(list(x1 = x1_1, y1 = y1_1, x2 = x1_2, y2 = y1_2),
list(x1 = x2_1, y1 = y2_1, x2 = x2_2, y2 = y2_2),
list(x1 = x3_1, y1 = y3_1, x2 = x3_2, y2 = y3_2)))
}
letter_y = function() {
t = seq(0, 1, by=delta_t)
n = length(t)
# Stroke 1:
x1_1 = seq(0.02, 0.5, length.out = n)
y1_1 = 0.4*t
x1_2 = x1_1
y1_2 = y1_1 + 0.1*sqrt(2)
# Stroke 2:
x2_1 = seq(0.5, 0.98, length.out = n)
y2_1 = y1_1[n:1]
x2_2 = x2_1
y2_2 = y1_2[n:1]
# Stroke 3:
x3_1 = 0.45 + 0*t
y3_1 = 1 - 0.5*t
x3_2 = 0.55 + 0*t
y3_2 = y3_1
return(list(list(x1 = x1_1, y1 = y1_1, x2 = x1_2, y2 = y1_2),
list(x1 = x2_1, y1 = y2_1, x2 = x2_2, y2 = y2_2),
list(x1 = x3_1, y1 = y3_1, x2 = x3_2, y2 = y3_2)))
}
letter_z = function() {
t = seq(0, 1, by=delta_t)
n = length(t)
# Stroke 1:
x1_1 = seq(0.02, 0.98, length.out = n)
y1_1 = 0*t
x1_2 = x1_1
y1_2 = 0.1 + 0*t
# Stroke 2:
x2_1 = seq(0.02, 0.98 -0.1*sqrt(2), length.out = n)
y2_1 = seq(0.9, 0.1, length.out = n)
x2_2 = seq(0.02 + 0.1*sqrt(2), 0.98, length.out = n)
y2_2 = y2_1
# Stroke 3:
x3_1 = x1_1
y3_1 = 0.9 + 0*t
x3_2 = x1_1
y3_2 = 1 + 0*t
return(list(list(x1 = x1_1, y1 = y1_1, x2 = x1_2, y2 = y1_2),
list(x1 = x2_1, y1 = y2_1, x2 = x2_2, y2 = y2_2),
list(x1 = x3_1, y1 = y3_1, x2 = x3_2, y2 = y3_2)))
}
letter_strokes = function(letter_str) {
switch(letter_str,
A = letter_a(),
B = letter_b(),
C = letter_c(),
D = letter_d(),
E = letter_e(),
F = letter_f(),
G = letter_g(),
H = letter_h(),
I = letter_i(),
J = letter_j(),
K = letter_k(),
L = letter_l(),
M = letter_m(),
N = letter_n(),
O = letter_o(),
P = letter_p(),
Q = letter_q(),
R = letter_r(),
S = letter_s(),
T = letter_t(),
U = letter_u(),
V = letter_v(),
W = letter_w(),
X = letter_x(),
Y = letter_y(),
Z = letter_z()
)
}
# Create the set of strokes needed to write the word.
num_chars = nchar(fractal_str)
char_width = 1/num_chars
letter_gap = (1 - letter_width)/2
for (i in 1:num_chars) {
this_char = substr(fractal_str, i, i)
this_strokes = letter_strokes(this_char)
start_x = (i - 1 + letter_gap)*char_width
end_x = (i - letter_gap)*char_width
for (j in 1:length(this_strokes)) {
this_strokes[[j]]$x1 = start_x + this_strokes[[j]]$x1*letter_width / num_chars
this_strokes[[j]]$x2 = start_x + this_strokes[[j]]$x2*letter_width / num_chars
}
if (i == 1) {
base_strokes = this_strokes
} else {
base_strokes = c(base_strokes, this_strokes)
}
}
assign("base_strokes", value=base_strokes, pos=globalenv())
regularise_stroke = function(x, y) {
# Function to convert a stroke into segments of equal length.
n = length(x)
xi = x[1:(n-1)]
yi = y[1:(n-1)]
xf = x[2:n]
yf = y[2:n]
arc_lengths = sqrt((xf - xi)^2 + (yf - yi)^2)
if (max(arc_lengths) - min(arc_lengths) > 1) {
total_length = sum(arc_lengths)
cumul_lengths = c(0, cumsum(arc_lengths))
delta_length = total_length / (n-1)
reg_cumul_lengths = seq(delta_length, total_length, by=delta_length)
indices_floor = sapply(reg_cumul_lengths, function(x) max(which(cumul_lengths < x)))
indices_ceiling = sapply(reg_cumul_lengths, function(x) min(which(cumul_lengths >= x)))
interp_dist = (reg_cumul_lengths - cumul_lengths[indices_floor]) / (cumul_lengths[indices_ceiling] - cumul_lengths[indices_floor])
reg_x = x
reg_y = y
reg_x[2:n] = x[indices_floor] + interp_dist * (x[indices_ceiling] - x[indices_floor])
reg_y[2:n] = y[indices_floor] + interp_dist * (y[indices_ceiling] - y[indices_floor])
return(list(x=reg_x, y=reg_y))
} else {
return(list(x=x, y=y))
}
}
pixel_points = function(stroke_x, stroke_y, x1, y1, x2, y2, iter_level) {
# Returns pixel locations of the stroke in the next iteration of the fractal.
# stroke_x and stroke_y are from base_strokes: they have to be
# transformed so that they fit inside the top and bottom of the
# current stroke, defined by (x1, y1) and (x2, y2).
#
# stroke_x defines how far along each curve to go; stroke_y says where
# to define the point in between the two curves.
#
# iter_level is only passed to this function because I was using it
# while debugging.
n = length(stroke_x)
t = seq(0, 1, by = delta_t)
indices_floor = sapply(stroke_x, function(x) max(which(t < x)))
indices_ceiling = sapply(stroke_x, function(x) min(which(t >= x)))
interp_dist = (stroke_x - t[indices_floor]) / (t[indices_ceiling] - t[indices_floor])
temp_x1 = x1[indices_floor] + interp_dist * (x1[indices_ceiling] - x1[indices_floor])
temp_y1 = y1[indices_floor] + interp_dist * (y1[indices_ceiling] - y1[indices_floor])
temp_x2 = x2[indices_floor] + interp_dist * (x2[indices_ceiling] - x2[indices_floor])
temp_y2 = y2[indices_floor] + interp_dist * (y2[indices_ceiling] - y2[indices_floor])
temp_x = temp_x1 + stroke_y * (temp_x2 - temp_x1)
temp_y = temp_y1 + stroke_y * (temp_y2 - temp_y1)
reg_stroke = regularise_stroke(temp_x, temp_y)
new_x = reg_stroke$x
new_y = reg_stroke$y
return(list(x = new_x, y = new_y))
}
write_fractal = function(word_str, x1, y1, x2, y2, iter_level) {
# (x1, y1): parametrised top of stroke
# (x2, y2): parametrised bottom of stroke
n = length(x1)
for (i in 1:length(base_strokes)) {
if (iter_level < 3) {
print(sprintf("Doing stroke %d at iteration %d", i, iter_level))
}
pixels1 = pixel_points(base_strokes[[i]]$x1, base_strokes[[i]]$y1, x1, y1, x2, y2, iter_level)
pixels2 = pixel_points(base_strokes[[i]]$x2, base_strokes[[i]]$y2, x1, y1, x2, y2, iter_level)
width = max(pixels1$x) - min(pixels1$x)
height = max(pixels1$y) - min(pixels1$y)
this_x0 = pixels1$x[1]
this_y0 = pixels1$y[2]
if ((width + height < 3) | (iter_level > max_iter)) {
fractal_img[floor(this_y0), floor(this_x0)] <<- 1
} else {
write_fractal(word_str, pixels1$x, pixels1$y, pixels2$x, pixels2$y, iter_level + 1)
}
}
return(0)
}
# Define initial "stroke" in which to start the iterations:
t = seq(0, 1, by=delta_t)
x1 = 1 + t*(M - 1)
x2 = x1
y1 = 1 + 0*t
y2 = N + 0*t
write_fractal(fractal_str, x1, y1, x2, y2, 1)
fractal_img <<- 1 - fractal_img
# Transpose because EBImage's convention is different to my head's:
fractal_img <<- t(fractal_img)
image_towrite = Image(fractal_img, colormode=Color)
writeImage(image_towrite, "fractal.png")
